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Hint for single sum problems: In every single sum future value and present value problem, there are 4 variables: FV, PV, i, and n When doing problems, you will be given 3 of these variables and asked to solve for the 4th variable. Keeping this in mind makes “time value” problems much easier!

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The Time Value of Money Compounding and Discounting Cash Flow Streams

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Annuities Annuity: a sequence of equal cash flows, occurring at the end of each period.

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Annuities

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Examples of Annuities: If you buy a bond, you will receive equal semi-annual coupon interest payments over the life of the bond. If you borrow money to buy a house or a car, you will pay a stream of equal payments.

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If you buy a bond, you will receive equal semi-annual coupon interest payments over the life of the bond. If you borrow money to buy a house or a car, you will pay a stream of equal payments. Examples of Annuities:

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Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?

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Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?

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Calculator Solution: P/Y = 1I = 8N = 3 PMT = -1,000 FV = $3, Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?

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Calculator Solution: P/Y = 1I = 8N = 3 PMT = -1,000 FV = $3, Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?

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Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?

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Mathematical Solution: Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?

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Mathematical Solution: FV = PMT (FVIFA i, n ) Future Value - annuity If you invest $1,000 each year at 8%, how much would you have after 3 years?

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Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?

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Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?

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Calculator Solution: P/Y = 1I = 8N = 3 PMT = -1,000 PV = $2, Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?

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Calculator Solution: P/Y = 1I = 8N = 3 PMT = -1,000 PV = $2, Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?

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Mathematical Solution: Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?

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Mathematical Solution: PV = PMT (PVIFA i, n ) Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?

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Mathematical Solution: PV = PMT (PVIFA i, n ) PV = 1,000 (PVIFA.08, 3 ) (use PVIFA table, or) Present Value - annuity What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?

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Perpetuities Suppose you will receive a fixed payment every period (month, year, etc.) forever. This is an example of a perpetuity. You can think of a perpetuity as an annuity that goes on forever.

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Present Value of a Perpetuity When we find the PV of an annuity, we think of the following relationship:

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Present Value of a Perpetuity When we find the PV of an annuity, we think of the following relationship: PV = PMT (PVIFA i, n ) PV = PMT (PVIFA i, n )

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Mathematically,

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(PVIFA i, n ) =

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Mathematically, (PVIFA i, n ) = (1 + i) n i

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Mathematically, (PVIFA i, n ) = We said that a perpetuity is an annuity where n = infinity. What happens to this formula when n gets very, very large? (1 + i) n i

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When n gets very large,

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1 - 1 (1 + i) n i

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When n gets very large, this becomes zero (1 + i) n i

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When n gets very large, this becomes zero. So we’re left with PVIFA = 1 i (1 + i) n i

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So, the PV of a perpetuity is very simple to find: Present Value of a Perpetuity

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PMT i PV = So, the PV of a perpetuity is very simple to find: Present Value of a Perpetuity

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What should you be willing to pay in order to receive $10,000 annually forever, if you require 8% per year on the investment?

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PMT $10,000 PMT $10,000 i.08 i.08 PV = =

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What should you be willing to pay in order to receive $10,000 annually forever, if you require 8% per year on the investment? PMT $10,000 PMT $10,000 i.08 i.08 = $125,000 PV = =

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Ordinary Annuity vs. Annuity Due $1000 $1000 $

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Begin Mode vs. End Mode

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Begin Mode vs. End Mode year year year 5 6 7

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Begin Mode vs. End Mode year year year PVinENDMode

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Begin Mode vs. End Mode year year year PVinENDModeFVinENDMode

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Begin Mode vs. End Mode year year year 6 7 8

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Begin Mode vs. End Mode year year year PVinBEGINMode

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Begin Mode vs. End Mode year year year PVinBEGINModeFVinBEGINMode

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Earlier, we examined this “ordinary” annuity:

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Earlier, we examined this “ordinary” annuity: Using an interest rate of 8%, we find that:

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Earlier, we examined this “ordinary” annuity: Using an interest rate of 8%, we find that: The Future Value (at 3) is $3,

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Earlier, we examined this “ordinary” annuity: Using an interest rate of 8%, we find that: The Future Value (at 3) is $3, The Present Value (at 0) is $2,

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What about this annuity? Same 3-year time line, Same 3 $1000 cash flows, but The cash flows occur at the beginning of each year, rather than at the end of each year. This is an “annuity due.”

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Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3?

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Calculator Solution: Mode = BEGIN P/Y = 1I = 8 N = 3 PMT = -1,000 FV = $3, Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3?

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Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? Calculator Solution: Mode = BEGIN P/Y = 1I = 8 N = 3 PMT = -1,000 FV = $3,506.11

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Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? Mathematical Solution: Simply compound the FV of the ordinary annuity one more period:

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Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: FV = PMT (FVIFA i, n ) (1 + i)

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Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: FV = PMT (FVIFA i, n ) (1 + i) FV = 1,000 (FVIFA.08, 3 ) (1.08) (use FVIFA table, or)

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Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: FV = PMT (FVIFA i, n ) (1 + i) FV = 1,000 (FVIFA.08, 3 ) (1.08) (use FVIFA table, or) FV = PMT (1 + i) n - 1 i (1 + i)

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Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: FV = PMT (FVIFA i, n ) (1 + i) FV = 1,000 (FVIFA.08, 3 ) (1.08) (use FVIFA table, or) FV = PMT (1 + i) n - 1 i FV = 1,000 (1.08) = $3, (1 + i) (1.08)

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Present Value - annuity due What is the PV of $1,000 at the beginning of each of the next 3 years, if your opportunity cost is 8%?

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Calculator Solution: Mode = BEGIN P/Y = 1I = 8 N = 3 PMT = 1,000 PV = $2, Present Value - annuity due What is the PV of $1,000 at the beginning of each of the next 3 years, if your opportunity cost is 8%?

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Calculator Solution: Mode = BEGIN P/Y = 1I = 8 N = 3 PMT = 1,000 PV = $2, Present Value - annuity due What is the PV of $1,000 at the beginning of each of the next 3 years, if your opportunity cost is 8%?

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Present Value - annuity due Mathematical Solution:

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Present Value - annuity due Mathematical Solution: Simply compound the FV of the ordinary annuity one more period:

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Annual Percentage Yield (APY) Which is the better loan: 8% compounded annually, or 7.85% compounded quarterly? We can’t compare these nominal (quoted) interest rates, because they don’t include the same number of compounding periods per year! We need to calculate the APY.

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Retirement Example If you invest $400 at the end of each month for the next 30 years at 12%, how much would you have at the end of year 30?

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Mathematical Solution:

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Retirement Example If you invest $400 at the end of each month for the next 30 years at 12%, how much would you have at the end of year 30? Mathematical Solution: FV = PMT (FVIFA i, n )

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Retirement Example If you invest $400 at the end of each month for the next 30 years at 12%, how much would you have at the end of year 30? Mathematical Solution: FV = PMT (FVIFA i, n ) FV = 400 (FVIFA.01, 360 ) (can’t use FVIFA table)

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Retirement Example If you invest $400 at the end of each month for the next 30 years at 12%, how much would you have at the end of year 30? Mathematical Solution: FV = PMT (FVIFA i, n ) FV = 400 (FVIFA.01, 360 ) (can’t use FVIFA table) FV = PMT (1 + i) n - 1 i

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Retirement Example If you invest $400 at the end of each month for the next 30 years at 12%, how much would you have at the end of year 30? Mathematical Solution: FV = PMT (FVIFA i, n ) FV = 400 (FVIFA.01, 360 ) (can’t use FVIFA table) FV = PMT (1 + i) n - 1 i FV = 400 (1.01) = $1,397,

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If you borrow $100,000 at 7% fixed interest for 30 years in order to buy a house, what will be your monthly house payment? House Payment Example

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If you borrow $100,000 at 7% fixed interest for 30 years in order to buy a house, what will be your monthly house payment?

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Team Assignment Upon retirement, your goal is to spend 5 years traveling around the world. To travel in style will require $250,000 per year at the beginning of each year. If you plan to retire in 30 years, what are the equal monthly payments necessary to achieve this goal? The funds in your retirement account will compound at 10% annually.

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How much do we need to have by the end of year 30 to finance the trip? PV 30 = PMT (PVIFA.10, 5 ) (1.10) = = 250,000 (3.7908) (1.10) = = $1,042,